Scenarios¶

Multi-year planning with uncertain demand scenarios.

This notebook introduces:

Periods: Multiple planning years with different conditions
Scenarios: Uncertain futures (mild vs. harsh winter)
Scenario weights: Probability-weighted optimization
Multi-dimensional data: Parameters that vary by time, period, and scenario

Setup¶

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import numpy as np
import pandas as pd
import plotly.express as px

import flixopt as fx

fx.CONFIG.notebook()
import numpy as np import pandas as pd import plotly.express as px import flixopt as fx fx.CONFIG.notebook()

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flixopt.config.CONFIG

The Planning Problem¶

We're designing a heating system with:

3 periods (years): 2024, 2025, 2026 - gas prices expected to rise
2 scenarios: "Mild Winter" (60% probability) and "Harsh Winter" (40% probability)
Investment decision: Size of CHP unit (made once, works across all futures)

The optimizer finds the investment that minimizes expected cost across all scenarios.

Define Dimensions¶

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# Time horizon: one representative winter week
timesteps = pd.date_range('2024-01-15', periods=168, freq='h')  # 7 days

# Planning periods (years)
periods = pd.Index([2024, 2025, 2026], name='period')

# Scenarios with probabilities
scenarios = pd.Index(['Mild Winter', 'Harsh Winter'], name='scenario')
scenario_weights = np.array([0.6, 0.4])  # 60% mild, 40% harsh

print(f'Time dimension: {len(timesteps)} hours')
print(f'Periods: {list(periods)}')
print(f'Scenarios: {list(scenarios)}')
print(f'Scenario weights: {dict(zip(scenarios, scenario_weights, strict=False))}')
# Time horizon: one representative winter week timesteps = pd.date_range('2024-01-15', periods=168, freq='h') # 7 days # Planning periods (years) periods = pd.Index([2024, 2025, 2026], name='period') # Scenarios with probabilities scenarios = pd.Index(['Mild Winter', 'Harsh Winter'], name='scenario') scenario_weights = np.array([0.6, 0.4]) # 60% mild, 40% harsh print(f'Time dimension: {len(timesteps)} hours') print(f'Periods: {list(periods)}') print(f'Scenarios: {list(scenarios)}') print(f'Scenario weights: {dict(zip(scenarios, scenario_weights, strict=False))}')

Time dimension: 168 hours
Periods: [2024, 2025, 2026]
Scenarios: ['Mild Winter', 'Harsh Winter']
Scenario weights: {'Mild Winter': np.float64(0.6), 'Harsh Winter': np.float64(0.4)}

Create Scenario-Dependent Demand Profiles¶

Heat demand differs significantly between mild and harsh winters:

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hours = np.arange(168)
hour_of_day = hours % 24

# Base daily pattern (kW): higher in morning/evening
daily_pattern = np.select(
    [
        (hour_of_day >= 6) & (hour_of_day < 9),  # Morning peak
        (hour_of_day >= 9) & (hour_of_day < 17),  # Daytime
        (hour_of_day >= 17) & (hour_of_day < 22),  # Evening peak
    ],
    [180, 120, 160],
    default=100,  # Night
).astype(float)

# Add random variation
np.random.seed(42)
noise = np.random.normal(0, 10, len(timesteps))

# Mild winter: lower demand
mild_demand = daily_pattern * 0.8 + noise
mild_demand = np.clip(mild_demand, 60, 200)

# Harsh winter: higher demand
harsh_demand = daily_pattern * 1.3 + noise * 1.5
harsh_demand = np.clip(harsh_demand, 100, 280)

# Create DataFrame with scenario columns (flixopt uses column names to match scenarios)
heat_demand = pd.DataFrame(
    {
        'Mild Winter': mild_demand,
        'Harsh Winter': harsh_demand,
    },
    index=timesteps,
)

print(f'Mild winter demand: {mild_demand.min():.0f} - {mild_demand.max():.0f} kW')
print(f'Harsh winter demand: {harsh_demand.min():.0f} - {harsh_demand.max():.0f} kW')
hours = np.arange(168) hour_of_day = hours % 24 # Base daily pattern (kW): higher in morning/evening daily_pattern = np.select( [ (hour_of_day >= 6) & (hour_of_day < 9), # Morning peak (hour_of_day >= 9) & (hour_of_day < 17), # Daytime (hour_of_day >= 17) & (hour_of_day < 22), # Evening peak ], [180, 120, 160], default=100, # Night ).astype(float) # Add random variation np.random.seed(42) noise = np.random.normal(0, 10, len(timesteps)) # Mild winter: lower demand mild_demand = daily_pattern * 0.8 + noise mild_demand = np.clip(mild_demand, 60, 200) # Harsh winter: higher demand harsh_demand = daily_pattern * 1.3 + noise * 1.5 harsh_demand = np.clip(harsh_demand, 100, 280) # Create DataFrame with scenario columns (flixopt uses column names to match scenarios) heat_demand = pd.DataFrame( { 'Mild Winter': mild_demand, 'Harsh Winter': harsh_demand, }, index=timesteps, ) print(f'Mild winter demand: {mild_demand.min():.0f} - {mild_demand.max():.0f} kW') print(f'Harsh winter demand: {harsh_demand.min():.0f} - {harsh_demand.max():.0f} kW')

Mild winter demand: 60 - 163 kW
Harsh winter demand: 100 - 262 kW

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# Visualize demand scenarios with plotly
fig = px.line(
    heat_demand.iloc[:48],
    title='Heat Demand by Scenario (First 2 Days)',
    labels={'index': 'Time', 'value': 'kW', 'variable': 'Scenario'},
)
fig.update_traces(mode='lines')
fig
# Visualize demand scenarios with plotly fig = px.line( heat_demand.iloc[:48], title='Heat Demand by Scenario (First 2 Days)', labels={'index': 'Time', 'value': 'kW', 'variable': 'Scenario'}, ) fig.update_traces(mode='lines') fig

Create Period-Dependent Prices¶

Energy prices change across planning years:

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# Gas prices by period (€/kWh) - expected to rise
gas_prices = np.array([0.06, 0.08, 0.10])  # 2024, 2025, 2026

# Electricity sell prices by period (€/kWh) - CHP revenue
elec_prices = np.array([0.28, 0.34, 0.43])  # Rising with gas

print('Gas prices by period:')
for period, price in zip(periods, gas_prices, strict=False):
    print(f'  {period}: {price:.2f} €/kWh')

print('\nElectricity sell prices by period:')
for period, price in zip(periods, elec_prices, strict=False):
    print(f'  {period}: {price:.2f} €/kWh')
# Gas prices by period (€/kWh) - expected to rise gas_prices = np.array([0.06, 0.08, 0.10]) # 2024, 2025, 2026 # Electricity sell prices by period (€/kWh) - CHP revenue elec_prices = np.array([0.28, 0.34, 0.43]) # Rising with gas print('Gas prices by period:') for period, price in zip(periods, gas_prices, strict=False): print(f' {period}: {price:.2f} €/kWh') print('\nElectricity sell prices by period:') for period, price in zip(periods, elec_prices, strict=False): print(f' {period}: {price:.2f} €/kWh')

Gas prices by period:
  2024: 0.06 €/kWh
  2025: 0.08 €/kWh
  2026: 0.10 €/kWh

Electricity sell prices by period:
  2024: 0.28 €/kWh
  2025: 0.34 €/kWh
  2026: 0.43 €/kWh

Build the Flow System¶

Initialize with all dimensions:

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flow_system = fx.FlowSystem(
    timesteps=timesteps,
    periods=periods,
    scenarios=scenarios,
    scenario_weights=scenario_weights,
)
flow_system.add_carriers(
    fx.Carrier('gas', '#3498db', 'kW'),
    fx.Carrier('electricity', '#f1c40f', 'kW'),
    fx.Carrier('heat', '#e74c3c', 'kW'),
)

print(flow_system)
flow_system = fx.FlowSystem( timesteps=timesteps, periods=periods, scenarios=scenarios, scenario_weights=scenario_weights, ) flow_system.add_carriers( fx.Carrier('gas', '#3498db', 'kW'), fx.Carrier('electricity', '#f1c40f', 'kW'), fx.Carrier('heat', '#e74c3c', 'kW'), ) print(flow_system)

FlowSystem
==========
Timesteps: 168 (Hour) [2024-01-15 to 2024-01-21]
Periods: 3 (2024, 2025, 2026)
Scenarios: 2 (Mild Winter, Harsh Winter)
Status: ⚠

Add Components¶

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flow_system.add_elements(
    # === Buses ===
    fx.Bus('Electricity', carrier='electricity'),
    fx.Bus('Heat', carrier='heat'),
    fx.Bus('Gas', carrier='gas'),
    # === Effects ===
    fx.Effect('costs', '€', 'Total Costs', is_standard=True, is_objective=True),
    # === Gas Supply (price varies by period) ===
    fx.Source(
        'GasGrid',
        outputs=[
            fx.Flow(
                'Gas',
                bus='Gas',
                size=1000,
                effects_per_flow_hour=gas_prices,  # Array = varies by period
            )
        ],
    ),
    # === CHP Unit (investment decision) ===
    fx.linear_converters.CHP(
        'CHP',
        electrical_efficiency=0.35,
        thermal_efficiency=0.50,
        electrical_flow=fx.Flow(
            'P_el',
            bus='Electricity',
            # Investment optimization: find optimal CHP size
            size=fx.InvestParameters(
                minimum_size=0,
                maximum_size=100,
                effects_of_investment_per_size={'costs': 50},  # 50 €/kW annualized
            ),
            relative_minimum=0.3,
        ),
        thermal_flow=fx.Flow('Q_th', bus='Heat'),
        fuel_flow=fx.Flow('Q_fuel', bus='Gas'),
    ),
    # === Gas Boiler (existing backup) ===
    fx.linear_converters.Boiler(
        'Boiler',
        thermal_efficiency=0.90,
        thermal_flow=fx.Flow('Q_th', bus='Heat', size=500),
        fuel_flow=fx.Flow('Q_fuel', bus='Gas'),
    ),
    # === Electricity Sales (revenue varies by period) ===
    fx.Sink(
        'ElecSales',
        inputs=[
            fx.Flow(
                'P_el',
                bus='Electricity',
                size=100,
                effects_per_flow_hour=-elec_prices,  # Negative = revenue
            )
        ],
    ),
    # === Heat Demand (varies by scenario) ===
    fx.Sink(
        'HeatDemand',
        inputs=[
            fx.Flow(
                'Q_th',
                bus='Heat',
                size=1,
                fixed_relative_profile=heat_demand,  # DataFrame with scenario columns
            )
        ],
    ),
)
flow_system.add_elements( # === Buses === fx.Bus('Electricity', carrier='electricity'), fx.Bus('Heat', carrier='heat'), fx.Bus('Gas', carrier='gas'), # === Effects === fx.Effect('costs', '€', 'Total Costs', is_standard=True, is_objective=True), # === Gas Supply (price varies by period) === fx.Source( 'GasGrid', outputs=[ fx.Flow( 'Gas', bus='Gas', size=1000, effects_per_flow_hour=gas_prices, # Array = varies by period ) ], ), # === CHP Unit (investment decision) === fx.linear_converters.CHP( 'CHP', electrical_efficiency=0.35, thermal_efficiency=0.50, electrical_flow=fx.Flow( 'P_el', bus='Electricity', # Investment optimization: find optimal CHP size size=fx.InvestParameters( minimum_size=0, maximum_size=100, effects_of_investment_per_size={'costs': 50}, # 50 €/kW annualized ), relative_minimum=0.3, ), thermal_flow=fx.Flow('Q_th', bus='Heat'), fuel_flow=fx.Flow('Q_fuel', bus='Gas'), ), # === Gas Boiler (existing backup) === fx.linear_converters.Boiler( 'Boiler', thermal_efficiency=0.90, thermal_flow=fx.Flow('Q_th', bus='Heat', size=500), fuel_flow=fx.Flow('Q_fuel', bus='Gas'), ), # === Electricity Sales (revenue varies by period) === fx.Sink( 'ElecSales', inputs=[ fx.Flow( 'P_el', bus='Electricity', size=100, effects_per_flow_hour=-elec_prices, # Negative = revenue ) ], ), # === Heat Demand (varies by scenario) === fx.Sink( 'HeatDemand', inputs=[ fx.Flow( 'Q_th', bus='Heat', size=1, fixed_relative_profile=heat_demand, # DataFrame with scenario columns ) ], ), )

Run Optimization¶

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flow_system.optimize(fx.solvers.HighsSolver(mip_gap=0.01));
flow_system.optimize(fx.solvers.HighsSolver(mip_gap=0.01));

2025-12-17 12:55:41.890 WARNING  │ ┌─ Flow CHP(P_el) has a relative_minimum of <xarray.DataArray 'CHP(P_el)|relative_minimum' (time: 168, period: 3,
                                 │ │                                                  scenario: 2)> Size: 8kB
                                 │ │  array([[[0.3, 0.3],
                                 │ │          [0.3, 0.3],
                                 │ │          [0.3, 0.3]],
                                 │ │  
                                 │ │         [[0.3, 0.3],
                                 │ │          [0.3, 0.3],
                                 │ │          [0.3, 0.3]],
                                 │ │  
                                 │ │         [[0.3, 0.3],
                                 │ │          [0.3, 0.3],
                                 │ │          [0.3, 0.3]],
                                 │ │  
                                 │ │         ...,
                                 │ │  
                                 │ │         [[0.3, 0.3],
                                 │ │          [0.3, 0.3],
                                 │ │          [0.3, 0.3]],
                                 │ │  
                                 │ │         [[0.3, 0.3],
                                 │ │          [0.3, 0.3],
                                 │ │          [0.3, 0.3]],
                                 │ │  
                                 │ │         [[0.3, 0.3],
                                 │ │          [0.3, 0.3],
                                 │ │          [0.3, 0.3]]], shape=(168, 3, 2))
                                 │ │  Coordinates:
                                 │ │    * time      (time) datetime64[ns] 1kB 2024-01-15 ... 2024-01-21T23:00:00
                                 │ │    * period    (period) int64 24B 2024 2025 2026
                                 │ └─   * scenario  (scenario) object 16B 'Mild Winter' 'Harsh Winter' and no status_parameters. This prevents the Flow from switching inactive (flow_rate = 0). Consider using status_parameters to allow the Flow to be switched active and inactive.

Writing constraints.:   0%|          | 0/30 [00:00<?, ?it/s]

Writing constraints.:  53%|█████▎    | 16/30 [00:00<00:00, 158.12it/s]

Writing constraints.: 100%|██████████| 30/30 [00:00<00:00, 157.02it/s]

Writing continuous variables.:   0%|          | 0/28 [00:00<?, ?it/s]

Writing continuous variables.: 100%|██████████| 28/28 [00:00<00:00, 556.56it/s]

Writing binary variables.:   0%|          | 0/1 [00:00<?, ?it/s]

Writing binary variables.: 100%|██████████| 1/1 [00:00<00:00, 503.10it/s]

Running HiGHS 1.12.0 (git hash: 755a8e0): Copyright (c) 2025 HiGHS under MIT licence terms
MIP linopy-problem-n4itunv5 has 12201 rows; 12198 cols; 36444 nonzeros; 6 integer variables (6 binary)
Coefficient ranges:
  Matrix  [1e-05, 1e+02]
  Cost    [4e-01, 6e-01]
  Bound   [1e+00, 1e+03]
  RHS     [0e+00, 0e+00]
Presolving model
3036 rows, 2025 cols, 6072 nonzeros  0s
1332 rows, 669 cols, 2664 nonzeros  0s
1326 rows, 666 cols, 2652 nonzeros  0s
Presolve reductions: rows 1326(-10875); columns 666(-11532); nonzeros 2652(-33792) 

Solving MIP model with:
   1326 rows
   666 cols (3 binary, 0 integer, 0 implied int., 663 continuous, 0 domain fixed)
   2652 nonzeros

Src: B => Branching; C => Central rounding; F => Feasibility pump; H => Heuristic;
     I => Shifting; J => Feasibility jump; L => Sub-MIP; P => Empty MIP; R => Randomized rounding;
     S => Solve LP; T => Evaluate node; U => Unbounded; X => User solution; Y => HiGHS solution;
     Z => ZI Round; l => Trivial lower; p => Trivial point; u => Trivial upper; z => Trivial zero

        Nodes      |    B&B Tree     |            Objective Bounds              |  Dynamic Constraints |       Work      
Src  Proc. InQueue |  Leaves   Expl. | BestBound       BestSol              Gap |   Cuts   InLp Confl. | LpIters     Time

 J       0       0         0   0.00%   -inf            5767.622795        Large        0      0      0         0     0.0s
         1       0         1 100.00%   5727.336398     5767.622795        0.70%        0      0      0       666     0.0s

Solving report
  Model             linopy-problem-n4itunv5
  Status            Optimal
  Primal bound      5767.62279485
  Dual bound        5727.33639803
  Gap               0.698% (tolerance: 1%)
  P-D integral      0.00883984914227
  Solution status   feasible
                    5767.62279485 (objective)
                    0 (bound viol.)
                    0 (int. viol.)
                    0 (row viol.)
  Timing            0.04
  Max sub-MIP depth 0
  Nodes             1
  Repair LPs        0
  LP iterations     666
                    0 (strong br.)
                    0 (separation)
                    0 (heuristics)

Analyze Results¶

Optimal Investment Decision¶

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chp_size = flow_system.statistics.sizes['CHP(P_el)']
total_cost = flow_system.solution['costs']

print('=== Investment Decision ===')
print(f'Optimal CHP size: {chp_size.round(1).to_pandas()} kW electrical')
print(f'Thermal capacity: {(chp_size * 0.50 / 0.35).round(1).to_pandas()} kW')
print(f'\nExpected total cost: {total_cost.round(2).to_pandas()} €')
chp_size = flow_system.statistics.sizes['CHP(P_el)'] total_cost = flow_system.solution['costs'] print('=== Investment Decision ===') print(f'Optimal CHP size: {chp_size.round(1).to_pandas()} kW electrical') print(f'Thermal capacity: {(chp_size * 0.50 / 0.35).round(1).to_pandas()} kW') print(f'\nExpected total cost: {total_cost.round(2).to_pandas()} €')

=== Investment Decision ===
Optimal CHP size: scenario  Mild Winter  Harsh Winter
period                             
2024              0.0           0.0
2025              0.0           0.0
2026              0.0           0.0 kW electrical
Thermal capacity: scenario  Mild Winter  Harsh Winter
period                             
2024              0.0           0.0
2025              0.0           0.0
2026              0.0           0.0 kW

Expected total cost: scenario  Mild Winter  Harsh Winter
period                             
2024          1153.37       1874.72
2025          1537.82       2499.62
2026          1922.28       3124.53 €

Heat Balance by Scenario¶

See how the system operates differently in each scenario:

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flow_system.statistics.plot.balance('Heat')
flow_system.statistics.plot.balance('Heat')

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CHP Operation Patterns¶

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flow_system.statistics.plot.heatmap('CHP(Q_th)')
flow_system.statistics.plot.heatmap('CHP(Q_th)')

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Multi-Dimensional Data Access¶

Results include all dimensions (time, period, scenario):

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# View dimensions
flow_rates = flow_system.statistics.flow_rates
print('Flow rates dimensions:', dict(flow_rates.sizes))

# Plot flow rates
flow_system.statistics.plot.flows()
# View dimensions flow_rates = flow_system.statistics.flow_rates print('Flow rates dimensions:', dict(flow_rates.sizes)) # Plot flow rates flow_system.statistics.plot.flows()

Flow rates dimensions: {'period': 3, 'scenario': 2, 'time': 169}

Out[12]:

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# CHP operation in harsh winter vs mild winter
chp_heat = flow_rates['CHP(Q_th)']

print('CHP Heat Output Statistics:')
for scenario in scenarios:
    scenario_data = chp_heat.sel(scenario=scenario)
    print(f'\n{scenario}:')
    for period in periods:
        period_data = scenario_data.sel(period=period)
        print(f'  {period}: avg={period_data.mean().item():.1f} kW, max={period_data.max().item():.1f} kW')
# CHP operation in harsh winter vs mild winter chp_heat = flow_rates['CHP(Q_th)'] print('CHP Heat Output Statistics:') for scenario in scenarios: scenario_data = chp_heat.sel(scenario=scenario) print(f'\n{scenario}:') for period in periods: period_data = scenario_data.sel(period=period) print(f' {period}: avg={period_data.mean().item():.1f} kW, max={period_data.max().item():.1f} kW')

CHP Heat Output Statistics:

Mild Winter:
  2024: avg=0.0 kW, max=0.0 kW
  2025: avg=0.0 kW, max=0.0 kW
  2026: avg=0.0 kW, max=0.0 kW

Harsh Winter:
  2024: avg=0.0 kW, max=0.0 kW
  2025: avg=0.0 kW, max=0.0 kW
  2026: avg=0.0 kW, max=0.0 kW

Sensitivity: What if Only Mild Winter?¶

Compare optimal CHP size if we only planned for mild winters:

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# Select only the mild winter scenario
fs_mild = flow_system.transform.sel(scenario='Mild Winter')
fs_mild.optimize(fx.solvers.HighsSolver(mip_gap=0.01))

chp_size_mild = fs_mild.statistics.sizes['CHP(P_el)']

print('=== Comparison ===')
print(f'CHP size (both scenarios): {chp_size.max("scenario").round(2).values} kW')
print(f'CHP size (mild only): {chp_size_mild.round(2).values} kW')
print(f'\nPlanning for uncertainty adds {(chp_size - chp_size_mild).round(2).values} kW capacity')
# Select only the mild winter scenario fs_mild = flow_system.transform.sel(scenario='Mild Winter') fs_mild.optimize(fx.solvers.HighsSolver(mip_gap=0.01)) chp_size_mild = fs_mild.statistics.sizes['CHP(P_el)'] print('=== Comparison ===') print(f'CHP size (both scenarios): {chp_size.max("scenario").round(2).values} kW') print(f'CHP size (mild only): {chp_size_mild.round(2).values} kW') print(f'\nPlanning for uncertainty adds {(chp_size - chp_size_mild).round(2).values} kW capacity')

2025-12-17 12:55:44.657 WARNING  │ ┌─ Flow CHP(P_el) has a relative_minimum of <xarray.DataArray 'CHP(P_el)|relative_minimum' (time: 168, period: 3)> Size: 4kB
                                 │ │  array([[0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │  ...
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3],
                                 │ │         [0.3, 0.3, 0.3]])
                                 │ │  Coordinates:
                                 │ │    * time      (time) datetime64[ns] 1kB 2024-01-15 ... 2024-01-21T23:00:00
                                 │ │    * period    (period) int64 24B 2024 2025 2026
                                 │ └─     scenario  <U11 44B 'Mild Winter' and no status_parameters. This prevents the Flow from switching inactive (flow_rate = 0). Consider using status_parameters to allow the Flow to be switched active and inactive.

Running HiGHS 1.12.0 (git hash: 755a8e0): Copyright (c) 2025 HiGHS under MIT licence terms
MIP linopy-problem-r02r_tqw has 6099 rows; 6099 cols; 18219 nonzeros; 3 integer variables (3 binary)
Coefficient ranges:
  Matrix  [1e-05, 1e+02]
  Cost    [1e+00, 1e+00]
  Bound   [1e+00, 1e+03]
  RHS     [0e+00, 0e+00]
Presolving model
1518 rows, 1014 cols, 3036 nonzeros  0s
948 rows, 477 cols, 1896 nonzeros  0s
948 rows, 477 cols, 1896 nonzeros  0s
Presolve reductions: rows 948(-5151); columns 477(-5622); nonzeros 1896(-16323) 

Solving MIP model with:
   948 rows
   477 cols (3 binary, 0 integer, 0 implied int., 474 continuous, 0 domain fixed)
   1896 nonzeros

Src: B => Branching; C => Central rounding; F => Feasibility pump; H => Heuristic;
     I => Shifting; J => Feasibility jump; L => Sub-MIP; P => Empty MIP; R => Randomized rounding;
     S => Solve LP; T => Evaluate node; U => Unbounded; X => User solution; Y => HiGHS solution;
     Z => ZI Round; l => Trivial lower; p => Trivial point; u => Trivial upper; z => Trivial zero

        Nodes      |    B&B Tree     |            Objective Bounds              |  Dynamic Constraints |       Work      
Src  Proc. InQueue |  Leaves   Expl. | BestBound       BestSol              Gap |   Cuts   InLp Confl. | LpIters     Time

 J       0       0         0   0.00%   -inf            4613.46344         Large        0      0      0         0     0.0s
         1       0         1 100.00%   4574.054792     4613.46344         0.85%        0      0      0       470     0.0s

Solving report
  Model             linopy-problem-r02r_tqw
  Status            Optimal
  Primal bound      4613.46344016
  Dual bound        4574.05479227
  Gap               0.854% (tolerance: 1%)
  P-D integral      0.00823189506654
  Solution status   feasible
                    4613.46344016 (objective)
                    0 (bound viol.)
                    0 (int. viol.)
                    0 (row viol.)
  Timing            0.02
  Max sub-MIP depth 0
  Nodes             1
  Repair LPs        0
  LP iterations     470
                    0 (strong br.)
                    0 (separation)
                    0 (heuristics)
=== Comparison ===
CHP size (both scenarios): [0. 0. 0.] kW
CHP size (mild only): [0. 0. 0.] kW

Planning for uncertainty adds [[0. 0.]
 [0. 0.]
 [0. 0.]] kW capacity

Energy Flow Sankey¶

A Sankey diagram visualizes the total energy flows through the system:

In [15]:

  Copied!     
 
flow_system.statistics.plot.sankey.flows()
flow_system.statistics.plot.sankey.flows()

Out[15]:

Key Concepts¶

Multi-Dimensional FlowSystem¶

flow_system = fx.FlowSystem(
    timesteps=timesteps,      # Time dimension
    periods=periods,          # Planning periods (years)
    scenarios=scenarios,      # Uncertain futures
    scenario_weights=weights, # Probabilities
)

Dimension-Varying Parameters¶

Data Shape	Meaning
Scalar	Same for all time/period/scenario
Array (n_periods,)	Varies by period
Array (n_scenarios,)	Varies by scenario
DataFrame with columns	Columns match scenario names
Full array (time, period, scenario)	Full specification

Scenario Optimization¶

The optimizer minimizes expected cost: $$\min \sum_s w_s \cdot \text{Cost}_s$$

where $w_s$ is the scenario weight (probability).

Selection Methods¶

# Select specific scenario
fs_mild = flow_system.transform.sel(scenario='Mild Winter')

# Select specific period
fs_2025 = flow_system.transform.sel(period=2025)

# Select time range
fs_day1 = flow_system.transform.sel(time=slice('2024-01-15', '2024-01-16'))

Summary¶

You learned how to:

Define multiple periods for multi-year planning
Create scenarios for uncertain futures
Use scenario weights for probability-weighted optimization
Pass dimension-varying parameters (arrays and DataFrames)
Select specific scenarios or periods for analysis

Next Steps¶

08a-Aggregation: Speed up large problems with resampling and clustering
08b-Rolling Horizon: Decompose large problems into sequential time segments